**The External Shapes of periodic 2-D arrays of repeated two-dimensional motifs **

**Rectangular Net.** (continued)

We continue our investigation concerning the faces, Forms and combinations of Forms in two-dimensional crystals. In the previous Part we had made a beginning with the study of the Crystal Class **m** of the 2-D Rectangular Crystal System (based on the Rectangular Net). We considered the first Plane Group, **Pm** belonging to that Class (Point Group).

Now we will consider its second Plane Group, **Pg**.

Figure 1. *Atomic aspects presented (to the growing environment) by a two-dimensional crystal of the Class ***m***. The point symmetry is indicated by the mirror line *(m)

The difference between the crystal of Figure 18 of Part Nine and the one depicted here -- as a difference in their

As for the position, as drawn, of the

Let us give Figure 18 of Part Nine again for easy comparison with the above Figure.

Figure 1a. *The mirror line, indicating the point symmetry of the crystal, will still be there when all translational symmetry is taken into account (i.e. when we look to it microscopically and detecting translations), in contrast to the case depicted in Figure 1. The difference between these two rectangular crystals -- which belong to the same Crystal Class, namely *

In explaining Figure 1 we stated

Figure 1b. *The motifs of the top and bottom faces of the 2-D crystal of Figure 1 are related by a glide line *(g)*, i.e. related by a translation followed by a reflection, as indicated. So the positions of the motifs associated with the top face are analogous to those of the bottom face except for a horizontal shift (according to the glide translation vector). If we undo this shift then the configuration of the motifs associated with the top face is symmetrical with respect to that of the motifs associated with the bottom face of the crystal. See Figures 1c and 1d.*

Figure 1c. *The horizontal translational difference of the upper and lower motif configurations is eliminated.*

Figure 1d. *The top and bottom faces are now symmetrical with respect to the glide line, which consequently becomes a mirror line.*

A possible face of a crystal of Class

Figure 2. *Atomic aspect as represented to the environment by the initial face depicted in Figure 11 of Part Nine.*

Figure 3. *Atomic aspect as presented to the environment by the implied face depicted in Figure 11 of Part Nine. Although the faces *

In any pattern representing Plane Group

Figure 4 shows the undoing of the just mentioned horizontal shift with respect to the pattern of Figure 3, and Figure 5 directly compares the result with the pattern of Figure 2.

Figure 4. *Undoing the horizontal shift (indicated by arrows), i.e. eliminating the translational component of the glide lines.*

Figure 5. *The result of the undoing of the horizontal shift, indicated in Figure 4, is given in the lower half of the present Figure. This then is directly compared with the pattern of Figure 2, which is (again) depicted in the upper half of the present Figure. We can clearly see that the two halves (top and bottom halves) are fully symmetric, meaning that the faces ***a*** and ***b*** are equivalent.*

Another orientation of a possible face effects yet another atomic aspect to the environment. See Figure 6.

Figure 6. *Atomic aspect of the initial face of Figure 13 in Part Nine -- in the present Figure indicated by *

Figure 7. *Atomic aspect of the implied face of Figure 13 in Part Nine -- in the present Figure indicated by *

When we shift the pattern of motifs of Figure 7 to the left by half a horizontal unit mesh dimension, i.e. when we eliminate the glide translational difference of this pattern with that of Figure 6, then we -- like in the previous case -- obtain two symmetrical halves, indicating that the faces

Figure 8. *After removing the glide translational difference between the pattern of Figure 6 and that of Figure 7, we can see that they are symmetrically related to each other with respect to a horizontal mirror line, expressing the point symmetry of the present Class ***m***.*

Now we will consider the atomic aspects in cases where we have to do with the third (and last) Plane Group belonging to the Point Group

The patterns representing this Plane Group are based on a rectangular lattice that has the same point symmetry as the rectangular lattices discussed so far, but is nevertheless fundamentally different. It is the

We will now consider the atomic aspects that are presented to the environment by the possible faces that we have derived earlier (in Part Nine), i.e. faces that can be constructed by the regular stacking of rectangular building blocks.

To begin with let us again give a pattern (of motifs) representing the Plane Group **Cm :**

Figure 9. *A pattern of periodically repeated motifs, representing Plane Group ***Cm***. The periodicity of the pattern is based on (i.e. described by) a centered rectangular net.*

Figure 10. *Atomic aspect presented by the faces of a 2-D crystal. The crystal consists of a combination of three Forms : one consisting of the top and bottom faces, a second consisting of the left face, and a third consistinf of the right face of the crystal.
The incomplete motifs at the faces express unsatisfied or distorted chemical valences.*

Other possible faces, presenting yet another atomic aspect to the environment, are given in the next Figure.

Figure 11. *Atomic aspect presented to the environment by the initial and implied faces of Figure 11 in Part Nine. In the present Figure these faces are indicated by ***a*** and ***b***. As can be seen, the whole pattern, and consequently also the face pair ***a***, ***b***, is symmetric with respect to a horizontal mirror line, expressing the point symmetry ***m*** of the pattern.*

Still other possible faces, presenting yet another atomic aspect to the environment, are depicted in Figure 12.

Figure 12. *Atomic aspect presented to the environment by the initial and implied faces of Figure 13 in Part Nine. In the present Figure these faces are indicated by ***c*** and ***d***. As can be seen, the whole pattern, and consequently also the face pair ***c***, ***d***, is symmetric with respect to a horizontal mirror line, expressing the point symmetry ***m*** of the pattern.*

We have now exhaustively considered the Class

Four Plane Groups belong to Class

- P2mm
- P2mg
- P2gg
- C2mm

The Class

We will now consider possible faces, Forms and combinations of Forms, belonging to our Class

Figure 13. *An introduced face, perpendicular to one of the crystallographic axes, implies a second face, because of either the action of the 2-fold rotation axis or of the mirror line parallel to it. The mirror line perpendicular to the introduced face has no effect on this face. The resulting face pair is a possible Form of the Class ***2mm***. It is an open Form, and can only exist in (2-D) crystals when it combines with other Forms of the Class such that a closed face configuration is formed.*

Figure 14. *An introduced face, perpendicular to the other of the crystallographic axes, also implies a second face, because of either the action of the 2-fold rotation axis or of the mirror line parallel to it. The mirror line perpendicular to the introduced face has no effect on this face. The resulting face pair is yet another possible Form of the Class ***2mm***. It is an open Form, and can only exist in (2-D) crystals when it combines with other Forms of the Class such that a closed face configuration is formed.*

These two Forms can enter into a combination with each other generating a 2-D crystal of the Class

Figure 15. *A 2-D crystal of the Class ***2mm***, resulting from a combination of two Forms (depicted in Figures 13 and 14), one consisting of the top and bottom faces, the other of the left and right faces. A crystal of the same shape we already met in the Class ***m*** (Figure 16 in Part Nine), but there the crystal was the result of the combination of three Forms : one consisting of the top and bottom faces, the second consisting of the left face, and the third consisting of the right face. Although the two crystals have the same shape they have different symmetries. They are elongated in the direction of the shortest unit mesh dimension because the faces growing in this direction cut through less lattice nodes than the other two faces of the crystal do, effecting them to be fast growing faces. They do not grow themselves out of existence because the other two faces are parallel to each other.*

Another Form can be derived from yet another initial face. See Figure 16.

Figure 16. *An introduced face is first duplicated by one of the mirror lines, and then the generated face pair is again duplicated by the other mirror plane. The 2-fold rotation axis does not add anything new. The resulting face configuration -- consisting of four faces -- is a ***closed*** Form, and can as such represent a 2-D crystal. Its shape is a rhombus. See the next two Figures for the final result.*

Figure 16a.

Figure 16b. *A two-dimensional crystal of the Class ***2mm*** of the 2-D Rectangular Crystal System. The crystal consists of one Form only.*

The Form of which the above crystal consists can still combine with other Forms. In the next Figures we show a combination of this Form with the Form of Figure 14.

Figure 17.

Figure 17a. *A two-dimensional crystal consisting of two Forms of the Class ***2mm***.*

Another Form can be generated from an initial face having yet another orientation.

See Figures 18 and 18a.

Figure 18. *An initial face is multiplied by the symmetry elements, resulting in a single Form having the shape of a rhombus. The final result is shown in the next Figure.*

Figure 18a. *A two-dimensional crystal of the Class ***2mm*** of the 2-D Rectangular Crystal System. The crystal consists of one Form only.*

Again another Form of this Class can be generated from an initial face having yet another orientation. See Figures 19 and 19a.

Figure 19. *An initial face is multiplied by the symmetry elements, resulting in a single Form having the shape of a rhombus. The final result is shown in the next Figure.*

Figure 19a. *A two-dimensional crystal of the Class ***2mm*** of the 2-D Rectangular Crystal System. The crystal consists of one Form only.*

In the next Part we look to the different

To continue, click HERE for Part Eleven.